Let `f(x)=["sinx"/x], x ne 0` , where [.] denotes the greatest integer function then `lim_(xto0)f(x)`

Options

A. does not exist
B. is equal to 1
C. is equal to 0
D. `lim_(xto0+)` f(x)=1

Answer

To solve the problem, we need to find the limit of the function \( f(x) = \left\lfloor \frac{\sin x}{x} \right\rfloor \) as \( x \) approaches 0, where \( \left\lfloor . \right\rfloor \) denotes the greatest integer function.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) \) is defined as the greatest integer less than or equal to \( \frac{\sin x}{x} \) for \( x \neq 0 \). We know from calculus that \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).

**Hint**: Recall the limit property of \( \frac{\sin x}{x} \) as \( x \) approaches 0.

2. **Behavior Near Zero**:
As \( x \) approaches 0, \( \frac{\sin x}{x} \) approaches 1. However, we need to consider the values of \( \frac{\sin x}{x} \) as \( x \) gets very close to 0 from both the left and the right.

**Hint**: Consider the values of \( \sin x \) and how it compares to \( x \) for small \( x \).

3. **Analyzing the Function**:
For \( x > 0 \), \( \sin x < x \), hence \( \frac{\sin x}{x} < 1 \). For \( x < 0 \), \( \sin x > x \), hence \( \frac{\sin x}{x} > 1 \). However, as \( x \) approaches 0 from either side, \( \frac{\sin x}{x} \) approaches 1.

**Hint**: Use the fact that \( \sin x \) is continuous and differentiable around 0.

4. **Applying the Greatest Integer Function**:
Since \( \frac{\sin x}{x} \) approaches 1 but is always less than 1 for \( x > 0 \), we have:
\[ 0 < \frac{\sin x}{x} < 1 \text{ for } x > 0 \]
Therefore, \( \left\lfloor \frac{\sin x}{x} \right\rfloor = 0 \) for \( x \) sufficiently close to 0 from the right.

**Hint**: Remember that the greatest integer function will take the largest integer less than or equal to the value.

5. **Limit Calculation**:
Now, we can conclude:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \left\lfloor \frac{\sin x}{x} \right\rfloor = 0 \]

**Hint**: Confirm that the limit from both sides (left and right) leads to the same conclusion.

### Final Answer:
Thus, the limit is:
\[ \lim_{x \to 0} f(x) = 0 \]